Finite mixtures in confirmatory factor-analysis models

In this paper, various types of finite mixtures of confirmatory factor-analysis models are proposed for handling data heterogeneity. Under the proposed mixture approach, observations are assumed to be drawn from mixtures of distinct confirmatory factor-analysis models. But each observation does not need to be identified to a particular model prior to model fitting. Several classes of mixture models are proposed. These models differ by their unique representations of data heterogeneity. Three different sampling schemes for these mixture models are distinguished. A mixed type of the these three sampling schemes is considered throughout this article. The proposed mixture approach reduces to regular multiple-group confirmatory factor-analysis under a restrictive sampling scheme, in which the structural equation model for each observation is assumed to be known. By assuming a mixture of multivariate normals for the data, maximum likelihood estimation using the EM (Expectation-Maximization) algorithm and the AS (Approximate-Scoring) method are developed, respectively. Some mixture models were fitted to a real data set for illustrating the application of the theory. Although the EM algorithm and the AS method gave similar sets of parameter estimates, the AS method was found computationally more efficient than the EM algorithm. Some comments on applying the mixture approach to structural equation modeling are made.Note: This paper is one of the Psychometric Society's 1995 Dissertation Award papers.—EditorThis article is based on the dissertation of the author. The author would like to thank Peter Bentler, who was the dissertation chair, for guidance and encouragement of this work. Eric Holman, Robert Jennrich, Bengt Muthén, and Thomas Wickens, who served as the committee members for the dissertation, had been very supportive and helpful. Michael Browne is appreciated for discussing some important points about the use of the approximate information in the dissertation. Thanks also go to an anonymous associate editor, whose comments were very useful for the revision of an earlier version of this article.     

Stochastic ordering using the latent trait and the sum score in polytomous IRT models

In a restricted class of item response theory (IRT) models for polytomous items the unweighted total score has monotone likelihood ratio (MLR) in the latent trait. MLR implies two stochastic ordering (SO) properties, denoted SOM and SOL, which are both weaker than MLR, but very useful for measurement with IRT models. Therefore, these SO properties are investigated for a broader class of IRT models for which the MLR property does not hold.In this study, first a taxonomy is given for nonparametric and parametric models for polytomous items based on the hierarchical relationship between the models. Next, it is investigated which models have the MLR property and which have the SO properties. It is shown that all models in the taxonomy possess the SOM property. However, counterexamples illustrate that many models do not, in general, possess the even more useful SOL property.Hemker's research was supported by the Netherlands Research Council, Grant 575-67-034. Junker's research was supported in part by the National Institutes of Health, Grant CA54852, and by the National Science Foundation, Grant DMS-94.04438.     

Inverses for Normal Modal Operators

Given a 1-ary sentence operator , we describe L - another 1-ary operator - as as a left inverse of in a given logic if in that logic every formula is provably equivalent to L. Similarly R is a right inverse of if is always provably equivalent to R. We investigate the behaviour of left and right inverses for taken as the operator of various normal modal logics, paying particular attention to the conditions under which these logics are conservatively extended by the addition of such inverses, as well as to the question of when, in such extensions, the inverses behave as normal modal operators in their own right.     

The Question of God in an Age of Science: Constructions of Reality and Ultimate Reality in Theology and Science

Both science and theology have lately faced a crisis of authority. Their shared realization of the extent to which knowledge is underdetermined by the data and socially constructed provides a kind of common ground for reconsideration of their respective methods of inquiry as well as of the status of the claims they have warrant to make. Both fields are now consciously and critically employing a models approach. This article proposes criteria for assessing models and applies the criteria to one model from each field. The model of understanding evolution as a struggle for existence is considered from the field of science, and the traditional model for understanding the God-world relation as that of a king's relation to his kingdom is considered from the field of theology. Each of these models is evaluated with respect to its credibility, religious viability, and moral adequacy. In each case an alternative analogy is proposed and argued for.     

Residualanalysis in the polytomous rasch model

Residuals for check of model fit in the polytomous Rasch model are examined. Comparisons are made between using counts for all response pattern and using item totals for score groups for the construction of the residuals. Comparisons are also, for the residuals based on score group totals, made between using as basis the item totals, or using the estimated item parameters. The developed methods are illustrated by two examples, one from a psychiatric rating scale, one from a Danish Welfare Study.     

A general linear model for estimating effect size in the presence of publication bias

When the process of publication favors studies with smallp-values, and hence large effect estimates, combined estimates from many studies may be biased. This paper describes a model for estimation of effect size when there is selection based on one-tailedp-values. The model employs the method of maximum likelihood in the context of a mixed (fixed and random) effects general linear model for effect sizes. It offers a test for the presence of publication bias, and corrected estimates of the parameters of the linear model for effect magnitude. The model is illustrated using a well-known data set on the benefits of psychotherapy.Authors' note: The contributions of the authors are considered equal, and the order of authorship was chosen to be reverse-alphabetical.     

An extension of the concept of specific objectivity

Comparisons of subjects are specifically objective if they do not depend on the items involved. Such comparisons are not restricted to the 1-parameter logistic latent trait model, but may also be defined within ordinal independence models and even within the 2-parameter logistic model.     

Calibration of sensory and cognitive judgments: A single model for both

In a recent issue of this journal, Winman and Juslin (34 , 135–148, 1993) present a model of the calibration of subjective probability judgments for sensory discrimination tasks. They claim that the model predicts a pervasive underconfidence bias observed in such tasks, and present evidence from a training experiment that they interpret as supporting the notion that different models are needed to describe judgment of confidence in sensory and in cognitive tasks. The model is actually part of the more comprehensive decision variable partition model of subjective probability calibration that was originally proposed in Ferrell and McGoey (Organizational Behavior and Human Performance, 26 , 32–53, 1980). The characteristics of the model are described and it is demonstrated that the model does not predict underconfidence, that it is fully compatible with the overconfidence frequently found in calibration studies with cognitive tasks, and that it well represents experimental results from such studies. It is concluded that only a single model is needed for both types of task.     

初中生友谊质量与孤独感的变化轨迹:一项多元潜变量增长模型分析

为考察初中生友谊质量与孤独感的变化轨迹及两者间的关系,采用关系网络问卷、孤独感量表对西安市的875名初中生（M=12.73±0.18;男生494名）进行三年追踪测量,使用多元潜变量增长模型进行分析。结果发现：（1）初中三年,中学生的友谊质量逐渐提高,孤独感水平逐渐下降;（2）初一时的友谊质量和孤独感水平存在显著的个体差异,初中三年期间二者的发展速度也存在个体差异;（3）初一时的友谊质量可显著预测初一时的孤独感水平（β=-1.77,p<0.001）及随后的发展速度（β=0.20,p<0.05）;（4）男生的孤独感水平初始值高于女生,友谊质量的初始值则低于女生;高SES个体孤独感的起始值高于低SES个体。本研究结果揭示了初中三年期间个体友谊质量与孤独感的动态变化关系及友谊质量对孤独感的预测作用,对改善青少年孤独感的现状有一定的现实指导意义。